Mathematics

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2007, No. 7



This volume contains papers submitted to the
International Conference “Algebra and Number Theory”
dedicated to the 80th anniversary of Prof. V.E.Voskresenskii
(Samara, Samara State University, May, 21-25, 2007)

 


V.E. Voskresenskii

SAMARA ALGEBRAIC AND GEOMETRIC ENCLAVE


N.A. Vavilov

CALCULATIONS IN EXCEPTIONAL GROUPS

Algebra and Number Theory, dedicated to the 80-th anniversary of Prof. V.E. Voskresenskii (Samara, 2007). We describe calculations in Chevalley groups G(Φ, R) over a commutative ring. We formulate recent results by A. Bak, R.Hazrat and the author on the nilpotent structure of K1(Φ, R), based on the method of localisation-completion. We thoroughly discuss geometric approaches to proof of main structure theorems, both the smcdecomposition of unipotents developed by A. Stepanov, the author and E.Plotkin since 1990, and the smcproof from the Book ≈ A2 — proof, discovered recently by the author, M.Gavrilovich and S.Nikolenko. We give also closely related results, such as the main lemma of the A3 - proof, proposed by the author to assault centrality of K2(Φ, R) for exceptional groups; a characterisation of the extended Chevalley group of type E6 over an arbitrary commutative ring, found recently by the author and A. Luzgarev; and finiteness of commutators in elementary generators, established by A. Stepanov and the author.


G.V. Voskresenskaya

THE SHIMURA SUMS FOR ARITHMETIC FUNCTIONS

In the paper the Shimura sums for arithmetic functions are studied. Several exact formulas are obtained and relations between the Shimura sums for various arithmetic functions are analyzed. Also a number of arithmetic identities are proved.


A.V. Grishin, L.M. Tsybulya

ON THE (p, n)-PROBLEM

In the paper the structure of the unitary closed T-space Wn generated by n-words in the relatively free algebra where k is an infinite field of characteristic p>0 and I is the T-ideal generated by the polynomial [[x1, x2], x3] is studied. An infinite irreducible system of generators is found in Wn. In this connection infinite chains of the in- finitely based T-subspaces in Wn are considered. The specific character of the case p = 2 is shown. Finally the multiplicative structure of the T-space Wn is studied.


V.G. Zhuravlev, Yu.Yu. Evseeva

ARITHMETIC PROPERTIES OF NATURAL NUMBER’ REPRESENTATIONS BY BINARY QUADRATIC FORMS

In the paper a review of the arithmetical properties of representations of natural numbers by binary positive defined quadratic forms is given. The cases of two-, three-, and four-class forms are considered. The forms with a cyclic class group are also discussed.


E.I. Konovalova

sl(3, C) DECOMPOSITION INTO A SUM OF TWO LIE SUBALGEBRAS

In this paper we obtain a classification of all decompositions of the Lie algebra sl(3, C) into a sum of two subspaces such that each of them is a Lie subalgebra. The classification is realized up to action of the group of automorphisms. The solution of formulated problem gives a classification of all solutions of MYBE for sl(3, C) represented as a subtraction of two projectors.


J.V. Kochetova

ON PROPERTIES OF IDEALS OF LATTICE-ORDERED LIE ALGEBRAS

In the paper the properties of partial-ordered Lie algebras over various fields are studied. Examples of directed Lie algebras, lattice-ordered Lie algebras and linearly-ordered Lie algebras are given. Results concerned with the properties of l-ideals of lattice-ordered Lie algebras over partial ordered fields are obtained.


V.V. Krasil’shchikov, A.V. Shutov

ASPECTS OF PUTTING LATTICES IN ONE-DIMENSIONAL QUASIPERIODICAL TILINGS

One-dimensional quasiperiodical tilings defined on the basis of the approach using irrational turns of a circle are considered in the paper. We obtaine a complete description of a wide class of progressions included in given tilings.


Yu.Yu. Krutickov

AFFINE REPRESENTATIONS OF 3-DIMENSIONAL ALGEBRAIC TORI

For an arbitrary three-dimensional algebraic k-torus T we found the minimal natural n(T) such that there exists a regular embedding of T to GLn(T),k . We describe this embedding obtain a realization of T as affine variety in .


L.N. Kurtova

ON A BINARY ADDITIVE PROBLEM WITH QUADRATIC FORMS

An analog of additive problem of divisors is considered. An asymptotic formula for the number of solutions of the equation involving quadratic forms is given.


A.V. Martynov

EXTREMAL NUMBERING 

The exact isoperimetric inequality for special graph is obtained.


S.I. Nebaluev, I.A. Klyaeva

POINTED TOLERANT CUBIC SINGULAR HOMOLOGY THEORY

In the paper several homological functors on category of tolerance spaces, which are useful in the theory of spectral sequences of fibre tolerance spaces are constructed.


A.N. Panov, V.V. Sevost’yanova

REGULAR N-ORBIT IN THE NILRADICAL OF A PARABOLIC SUBALGEBRA

In the paper the adjoint action of the triangular group in the nilradical of a parabolic subalgebra is studied. We set up general conjectures on the construction of field of invariants and the structure of orbits of maximal dimension. The conjectures is proved for parabolic subalgebras of a special type.


O.I. Cherevatenko

THE VARIETY OF LEIBNIZ ALGEBRAS DEFINED BY THE IDENTITY x(yz)t≡0

In this paper the variety of Leibniz algebras C defined by the identity x(yz)t≡0 over a field of characteristic 0 is studied. A description of some properties of the variety C is given. As a result we prove that if the condition   holds, the variety V is nilpotent, where B is the variety of Leibniz algebras defined by the identity x(yz)≡0.


A.V. Shutov

OPTIMUM ESTIMATES IN THE PROBLEM OF THE DISTRIBUTION OF FRACTIONAL PARTS OF THE SEQUENCE

In the paper a remainder term in the problem of the distribution of fractional parts of the sequence nα with irrational α is considered. In the case of bounded remainder interval we prove new estimates of this remainder term. The estimates are exact with respect to the order.


V. Garbaliauskieè

ON L-FUNCTION OF ELLIPTIC CURVES

In the paper a survey on universality theorems for L-functions of elliptic curves over the field of rational numbers as well as their derivatives is presented.


J. Genys

VALUE DISTRIBUTION OF GENERAL DIRICHLET SERIES

In the paper a survey of some probabilistic results and their application to universality for general Dirichlet series is given.


B. Kunyavskiĭ

ALGEBRAIC TORI — THIRTY YEARS AFTER

This paper is an extended version of my talk given at the International Conference ”Algebra and Number Theory” dedicated to the 80th anniversary of Prof. V.E.Voskresenskiĭ, which was held at the Samara State University in May 2007. The goal is to give an overview of results by V.E. Voskresenskiĭ on arithmetic and birational properties of algebraic tori which culminated in his monograph published 30 years ago. We put these results and ideas into somehow broader context and also to give a brief digest of the relevant activity related to the period after the English version of the monograph has been published.


A. Laurinčikas

VALUE DISTRIBUTION THEOREMS FOR THE ESTERMANN ZETA-FUNCTION

In the paper, a survey of mean-value estimates, zero distribution, universality and limit theorems in the sense of weak convergence of probability measures for the Estermann zeta-function is presented.


R. Macaitienè

THE DISCRETE UNIVERSALITY OF THE PERIODIC HURWITZ ZETA-FUNCTION

The paper contains a survey on continuous and discrete universality theorems for periodic zeta-functions. A sketch of the proof in the case of the discrete universality for the periodic Hurwitz zeta-function is given. Also, joint generalizations for periodic Hurwitz zeta-functions are formulated.